The present invention relates generally to an induction logging system for measuring characteristics of formations through which a wellbore has been or is being drilled. More particularly, the present invention relates to a method for processing induction logging data efficiently so that a useful amount of information can be obtained with a relatively small number of calculations. Still more particularly, the present invention relates to a method for interpolating real data efficiently using a physically appropriate response function as the basis function for the interpolation.
Modern petroleum drilling and production operations demand a great quantity of information relating to parameters and conditions downhole. Such information typically includes characteristics of the earth formations traversed by the wellbore, in addition to data relating to the size and configuration of the borehole itself. The collection of information relating to conditions downhole, which commonly is referred to as "logging," can be performed by several methods. Oil well logging has been known in the industry for many years as a technique for providing information to a driller regarding the particular earth formation being drilled. In conventional oil well wireline logging, a probe or "sonde" is lowered into the borehole after some or all of the well has been drilled, and is used to determine certain characteristics of the formations traversed by the borehole. The sonde may include one or more sensors to measure parameters downhole and typically is constructed as a hermetically sealed cylinder for housing the sensors, which hangs at the end of a long cable or "wireline." The cable or wireline provides mechanical support to the sonde and also provides an electrical connection between the sensors and associated instrumentation within the sonde, and electrical equipment located at the surface of the well. Normally, the cable supplies operating power to the sonde and is used as an electrical conductor to transmit information signals from the sonde to the surface. In accordance with conventional techniques, various parameters of the earth's formations are measured and correlated with the position of the sonde in the borehole, as the sonde is pulled uphole.
The sensors used in a wireline sonde usually include a source device for transmitting energy into the formation, and one or more receivers for detecting the energy reflected from the formation. Various sensors have been used to determine particular characteristics of the formation, including nuclear sensors, acoustic sensors, and electrical sensors. See generally J. Lab, A Practical Introduction to Borehole Geophysics (Society of Exploration Geophysicists 1986); D. R. Skinner, Introduction to Petroleum Production, Volume 1, at 54-63 (Gulf Publishing Co. 1981).
For a formation to contain petroleum, and for the formation to permit the petroleum to flow through it, the rock comprising the formation must have certain well known physical characteristics. One measurable characteristic is the resistivity (or conductivity) of the formation, which can be determined by inducing an alternating electromagnetic field into the formation by a transmitter/receiver coil arrangement. The electromagnetic field induces alternating electric (or eddy) currents in the formation in paths that are substantially coaxial with the transmitter. These currents in turn create a secondary electromagnetic field in the medium, inducing an alternating voltage at the receiver coil. If the current in the transmitter coil is kept constant, the eddy current intensity is proportional to the conductivity of the formation. Consequently, the conductivity of the formation determines the intensity of the secondary electromagnetic field, and thus, the amplitude of the voltage at the receiver coil. As will be apparent to one skilled in the art, the propagating electromagnetic wave suffers both attenuation and phase shift as it traverses the formation. See generally, James R. Jordan, et al., Well Logging II--Electric And Acoustic Logging, SPE Monograph Series, Volume 10, at 71-87 (1986).
An exemplary induction tool is shown in the prior art drawing of FIG. 1, in which one or more transmitters (T) and a plurality of receivers (R.sub.i) are shown in a logging sonde. An oscillator supplies alternating current to the transmitter coils, thereby inducing current in the receiver coils that is measured, processed and sent to the surface of the well for recording. The voltage induced in the receiver coils results from the sum of all eddy currents induced in the surrounding formations by all transmitters. Typically, phase sensitive detectors are used to measure those components of the receiver voltage that are in-phase or in quadrature with the transmitter current.
As noted, the induced eddy currents tend to flow in circular paths that are coaxial with the transmitter coil. As shown in FIG. 1, for a vertical borehole traversing horizontally layered formations, there is a general symmetry for the induced current around the logging tool. In this ideal situation, each line of current flow remains in the same formation along its entire flow path, and never crosses a bed boundary. In this type of layered environment, the presence of the borehole can, to a first approximation, be neglected and the tool response can be modeled as an array of coils immersed in the formation without a borehole. This approximation breaks down when the borehole fluid is highly conductive, however, the present invention is directed primarily to logging situations in which a non-conductive fluid is used.
Because the data measured by the sonde is influenced by several factors, the conductivity measurements typically require adjustment to compensate for these influences. One common approach for verifying data is to calculate an estimate of the formation conductivity on the basis of other, known factors, use computer modeling programs to generate a simulation of the measured conductivity, hereinafter referred to as the "apparent conductivity," and then compare the apparent conductivity to the measured values. The algorithms for these analyses are known in the art and are set out in the report Modeling Induction Tools in Layered Formations, G. A. Merchant, Halliburton Logging Services, given at the SPWLA Topical Conference, Apr. 22, 21-24, 1997 South Padre Island, Tex., which is hereby incorporated by reference in its entirety.
One method for estimating conductivity, for example, entails the use of geometrical factors. The Doll geometrical factor is the response of the induction tool to very thin beds in a low conductivity (high resistivity) medium. The Doll geometrical factor is named after H. D. Doll, who introduced it in a paper entitled "Introduction to Induction Logging and Application to Logging of Wells Drilled with Oil-based Mud," Petroleum Transactions, AIME, vol. 1, no. 6, June 1949, pp. 148-162. The Doll geometrical factor is also discussed by Moran and Kunz in "Basic Theory on Induction Logging and Application to Study of Two Coil Sonde," Geophysics, vol. XXVII, no. 6, December 1962, pp. 829-858, 26.
Heretofore, it has been necessary to calculate the apparent conductivity using complex and time-consuming iterations. Because each data point requires a lengthy series of calculations and because of the large number of data points, a system for reducing the number of calculations that must be performed is desired. Examples of the necessary calculations and previous attempts to streamline them are discussed in the paper cited above.